Self adjoint Puns

Hello everyone!

I'm gearing up for teaching in person (in Florida) next week, after nearly 2 years of teaching online. This is amid the worst numbers we have ever seen for COVID here, and Florida is one of the three hot spots of the world right now. It's a bit daunting, but I'm doing my best to leverage online platforms to run a flipped classroom and record everything to help students who are concerned and want to distance themselves.

But I'm not here to talk about any of that.

If you're only vaguely aware of Hilbert space theory and Operators, then the video linked below is a start on discussing the Spectral Theorem for Operators, where we go over the theory for Self Adjoint Compact Operators (following Lang's Real and Functional Analysis). I am motivating it with Dynamic Mode Decompositions, where we absolutely need this theory to establish convergence of models obtained from DMD algorithms to those of the true dynamics.

This is continuing my course on Data Driven Methods in Dynamical Systems that I started in spring, and this series expands on our discussion of Dynamic Mode Decompositions. The video linked below is setting us up to talk about how to get convergent routines, and it comes down to some 101 theorems from Functional Analysis.

Right now, the literature has settled on purely heuristic motivations for DMD, where convergence theories haven't been strong enough to get the convergence of the spectra in DMD. Some recent work of me and my colleagues have begun to illuminate how you can achieve actual convergence, and this video series is intended to build up to that new theory.

https://youtu.be/Bqz73NQ5XaQ

https://preview.redd.it/njhmlea52cs61.png?width=885&format=png&auto=webp&s=419989ed1facfa2581327182f2a108b1b0181797

https://preview.redd.it/ps845eb02cs61.png?width=792&format=png&auto=webp&s=84484506a029bf0cbd3d3ab47f747fd2a0740cfd

Hello

Does anyone have some example for a bounded self-adjoint operator or an application of the spectral theorem for bounded self-adjoint operators?

Right now I can only find self-adjoin linear operator on a finite dimensional Hilbert space, self-adjoint compact operators, multiplicate operator and the discrete laplacian operator on l²(Z).

(z₁Ψ+z₂Φ) *=(z₁ *)Ψ *+(z₂ *)Φ * for some z₁,z₂ ∈ ℂ.

(the asterisks on the complex numbers denote complex conjugation and the asterisks on the operators denote taking the adjoint)

Suppose T is a self-adjoint positve unbounded operator on a Hilbert space. I know there exists T^(1/2) self-adjoint and positive such that T=T^(1/2) T^(1/2), but I also know that I can define T^(1/2) via spectral theorem. Do the two definition give the same operator? If yes, why?

Thanks to anyone who'll answer.

Hi everyone, I'm trying to prove that any self-adjoint linear operator A:H->H on a Hilbert space is bounded. I figured the best way would be to show that A is continuous, and therefore bounded. I followed a path of the form:

take x,y close to eachother.

|A(x-y)|^2 = <A(x-y),A(x-y)> = <(x-y),(A^2)(x-y)> <= |x-y||A^2(x-y)|

But this didn't get me anywhere. Any tips would be appreciated

Any cool consequences? Any particular reasons why it's so cool?

In my physics course I learned that self-adjoint operators only have real eigenvalues and even proved it. When we take the momentum operator (we also proved that it is self-adjoint) p_x=-ihd/dx, e^x is a eigenfunction with the eigenvalue -ih, which is complex. I know that this can not be possible, but I don't know what exactly is wrong.

I'm gearing up for my Quals and need to make sure I understand the difference in case it gets asked. Would you guys agree with this distinction? An operator A is hermitian if <Au,v>=<u,Av> for all u,v in the domain of A. This doesn't necessarily mean A=A* as the domain of A* could be larger than the domain of A. So if A is hermitian and D[A]=D[A*] then A is self-adjoint Please correct me in even the smallest detail as I would much rather hear how wrong I am from you guys then from my qualifying committee

EDIT: formatting

OK.

I know that self-adjoint complex operators always have real eigenvalues. So measurements will always produce a real number which is desirable.

Also Stone's theorem. https://en.wikipedia.org/wiki/Stone%27s_theorem_on_one-parameter_unitar

But this doesn't quite add up to an intuitive understanding yet.

Hi guys,

I am currently working on a write up for my project on Hydrodynamic stability. I am currently proving Howard's semi-circle theorem using Drazin and Reid's book as a guide (as well as several online sources). Towards the start of the proof there is talk of taking the adjoint form of Rayleigh's Stability equation (RSE) and I'm not too sure what this means. Also, why is it not self-adjoint?

The next line then states we can take the self-adjoin form of the adjoint form of RSE... now I am beyond confused! Perhaps I am just completely misunderstanding this step, but any help would be massively appreciated.

Thanks

Given a self-adjoint matrix H, can I find a unitary transformation U s.t. UHU^-1 is a matrix with entries either being 0 or some real constant c (the same constant for each entry), not necessarily diagonal. Any ideas where I can look, or some easy counter example?

I have a self-adjoint differential operator (w'''' + n^2 * w'' = 0) with self-adjoint boundary conditions (w(0) = w'(0) = w(1) = w'(1) = 0). If my knowledge of the theory serves me right, this is a Hermitian operator and, thus, the eigenfunctions corresponding to different eigenvalues should be orthogonal.

The eigenfunctions can be divided into two groups: Even (n_i = 2π, 4π, 6π, ...) where w_i(x) = A_i*(1 - cos(n_i *x))

Odd (n_i = 2.86π, 4.92π, 6.94π, ...) where w_i(x) = A_i*(1 - cos(n_i x) - (2/n_i)(n_i *x - sin(n_i *x))

Orthogonality holds between an even and an odd eigenfunction, but not between two even or two odd. Am I incorrect in assuming orthogonality is guaranteed or is something else at play?

Let V be a finite-dimensional vector space over C, together with a Hermitian inner product < , >. Let T:V->V be a linear function. Prove that T is self-adjoint if <Tv,v> (- R

I know they generalize the conjugate transpose of a matrix but I am not sure how. I also know they involve a restriction mapping. I have seen some examples of them but I don't understand why the examples satisfy the definition.

Given a finite-dimensional vector space V over the complex numbers, with inner product <,> and A a linear map from V to V, prove that <Ax,x> is real for all x if and only if A is self-adjoint.

The reverse direction is easy: if A is self-adjoint, then [;\langle Ax,x \rangle = \overline{\langle x,Ax\rangle} = \overline{\langle A^*x,x\rangle} = \overline{\langle Ax,x\rangle};], where the overline denotes the complex conjugate.

It's the forward direction that gives me trouble. <Ax,x> is always real implies that [;\langle Ax,x \rangle = \overline{\langle Ax,x \rangle} = \langle x,Ax \rangle = \langle A^*x,x \rangle ;]. I suspect that this does imply that A=A^* , because <Ax,y>=<x,A^* y> characterizes the adjoint; maybe I don't see the obvious connection, but I'd like to show that implication more directly, if possible. Can anyone help me?

I just finished an introductory course to partial differential equations. We covered how to solve homogeneous problems by separation of variables and nonhomogeneous problems with the method of eigenfunction expansion.

At the heart of the course was solving the Sturm Liouville (S-L) eigenvalue problem to get your eigenvalues and eigenfunctions from the boundary conditions.

All semester when we solved the S-L problem we had to show that the differential operator and the boundary conditions made the operator self-adjoint, which I mastered, but the professor never really explained what exactly that tells us about the problem.

So my question is what is self-adjointness and why is it useful/important?

Suppose a linear transformation Q on a finite-dimensional real inner-product space is non-singular, self-adjoint and positive-definite.

I have shown that its eigenvalues are positive.

How may I show that there exists a positive, self-adjoint LT S such that S^2 = Q ?

Hello Everyone,

I created a youTube channel (here's the link) a few months ago in which I post detailed lectures in higher mathematics.

I have been uploading Real Analysis and Linear Algebra videos.

I have covered the following topics so far:

• Linear Algebra

1. Fields, vector spaces, bases, dimension.
2. Linear Maps, Rank-Nullity.
3. Connection with matrices and linear equations.
4. Dual spaces, annihilators and transpose.
5. Eigenvalues and minimal polynomial.
6. Primary decomposition, upper triangulability, diagonalizabilility.
7. Bilinear Forms, orthogonality, inner products, adjoint.
8. Real and complex spectral theorem.

• Real Analysis.

1. What are rational and real numbers?
2. Special subsets of real line, Bolzano-Weierstrass theorem, Heine-Borel Theorem.
3. Continuity.
4. Differentiation, mean value theorems etc.
5. Riemann integration.

Future lectures will cover

1. Application of integration (Taylor's theorem).
2. Infinite series.
3. Power series.
4. Special functions like the exponential and the logarithmic functions.
5. Any more relevant topics with sufficient interest from the audience.

The course will be complete by the end of February after which I plan to start with group theory.

Almost every lecture begins with two or three problems.

My aim behind making this videos is to write a video book so that any one who wants to learn need not look elsewhere (though, of course, other sources can surely help).

I hope that the people here would find the content useful and interesting.

Thank you.

PS. According to the forum rules self-prmotion on Saturdays are allowed so I hope I am not crossing any boundaries.

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

Do your worst!

I'm surprised it hasn't decade.

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

It really does, I swear!

Buenosdillas

They’re on standbi

Pilot on me!!